$$\int ax^n\text dx$$
let$$ax^n=u$$$$x=\left(\frac{u}{a}\right)^{1/n}=\frac{u^{1/n}}{a^{1/n}}$$$$\text dx=\frac{u^{1/n-1}}{na^{1/n}}\text du$$
$$\int ax^n\text dx\longrightarrow\frac{1}{na^{1/n}}\int u\cdot{u^{1/n-1}}\text du$$
$$=\frac{1}{na^{1/n}}\int {u^{1/n}}\text du$$$$=\frac{1}{na^{1/n}}\frac{u^{1/n+1}}{1/n+1}+c$$$$={}\frac{a^{1+n}x^{1+n}}{a^{1/n}+na^{1/n}}+c$$$$=\frac{ax^{1+n}}{1+n}+c$$

Question

$$\int ax^n\text dx$$
let$$ax^n=u$$$$x=\left(\frac{u}{a}\right)^{1/n}=\frac{u^{1/n}}{a^{1/n}}$$$$\text dx=\frac{u^{1/n-1}}{na^{1/n}}\text du$$
$$\int ax^n\text dx\longrightarrow\frac{1}{na^{1/n}}\int u\cdot{u^{1/n-1}}\text du$$
$$=\frac{1}{na^{1/n}}\int {u^{1/n}}\text du$$$$=\frac{1}{na^{1/n}}\frac{u^{1/n+1}}{1/n+1}+c$$$$={}\frac{a^{1+n}x^{1+n}}{a^{1/n}+na^{1/n}}+c$$$$=\frac{ax^{1+n}}{1+n}+c$$

unklerhaukus · Answer

what can i do with this

anonymous · Answer

integrate stuff ?
for example  $$\int\limits^{}3x^{2}=x^{3}+C$$

anonymous · Answer

Why would you integrate $ax^n$ like that... Just use the linearity of integrals and the "power rule".

unklerhaukus · Answer

i know right this is a one step problem 
, i realized this when i got the the end,

is there any use you can think for all this latex?

mathslover · Answer

your question is to integrate ax^n dx  ?

mathmate · Answer

Perhaps in recognizing terms in summing series.

unklerhaukus · Answer

pardon @mathmate ?

mathmate · Answer

What I mean is, when we sum series, sometimes it is helpful to recognize terms in different forms.  If each term is the integral / derivative of the terms of an expansion of a particular function, then the sum of the series is the integral/derivative of that function.

anonymous · Answer

Is Suggestion which is a little vague accepted ?

anonymous · Answer

@UnkleRhaukus  ?

unklerhaukus · Answer

?

anonymous · Answer

You are applying an holomorphic function to the variable - lets extend it to complex domain:

anonymous · Answer

$$z^{\frac{1}{n}} = e^{\frac{1}{n}\log z} $$

unklerhaukus · Answer

ok , not too fast now

anonymous · Answer

So NOW you are integrating on a differently-shaped domain in the complex plane - therefore you can do a lot of integrals differently - not this "naked" function alone but, for example , multiplied by some other function which looks simpler in the new representation

anonymous · Answer

These geometric- holomorphic transforms are widely used in some areas of science and engineering

anonymous · Answer

Look up antenna theory, Greens functions [ in many areas - wave propagation, solid state, QM, Field theory etc]

anonymous · Answer

Mr champion ?

unklerhaukus · Answer

go on

anonymous · Answer

That all seems consistent, but quite unnecessary.

anonymous · Answer

I appologised from the start -" Suggestion which is a little vague ".
I saw quite MASSIVE use of such domain transformation in some calculation in electromagnetism, and Wave propagation, and Green's functions computing generally. But it was some years ago, and I don't recall more precise area.
Look up "geometric transformations of the complex plane" and Green function/antenna theory/etc

anonymous · Answer

@UnkleRhaukus  ghmm .....

anonymous · Answer

ghhm Ghhm GHMM-MM-MM !

unklerhaukus · Answer

complexification

anonymous · Answer

what about some (purely symbolic) gratification ...

unklerhaukus · Answer

im so confused

anonymous · Answer

*is also confused*
What is going on here?
$$\int\limits ax^n dx = a \int\limits x^n dx = a \cdot \frac{x^{n+1}}{n+1} +c.$$
by power rule, so what's the issue?

anonymous · Answer

He showed how to compute it by a change of variables and asked whether it can be interesting/useful . I gave him the interpretation where it is used and he (good FREE lesson on human nature...) did NOT thank me...

anonymous · Answer

Another good free lesson on human nature is how often you outright ask for medals and fans  . . . ;-)